Answers to Supplementary Exercises

Answers to Survival Models exercises

  1. \(\quad\) (a) This function does satisfy the conditions for a survival function: it can be demonstrated that it is non-increasing (by proving that \(f(b) \leq f(a)\), \(\forall b>a\))

    \(\quad\) (b) \(s(0)=1\) and \(s(100)=0\); and continuous (by proving that \(\lim_{x\to c} s(x)=s(c)\); (b) \(\omega = 100\))

    \(\quad\) (c) \(0.76375\); (d) \(0.23196\)

  2. \(\quad\) (a) \(0.75\)

    \(\quad\) (b) \(0.27\)

    \(\quad\) \(0.68235\)

    \(\quad\) (d) \(0.5625\)

  3. \(\quad\) (d) \(\omega = 125\)

  4. \(\quad\) (a) This function does satisfy the conditions for a survival function: it can be demonstrated that it is non-increasing (by proving that \(f(b) \leq f(a)\), \(\forall b>a\)); \(s(0)=1\) and \(\lim_{x\to \infty} s(x)=0\); and continuous (by proving that \(\lim_{x\to c} s(x)=s(c)\))

  5. \(\quad\) (a) \(0.5\)

    \(\quad\) (b) \(0.3466\)

    \(\quad\) (c) \(0\)

    \(quad\) (d) \(0.5\)

  6. \(\quad 14\)

  7. \(\quad 16.36\)

  8. \(\quad 0.18182\)

  9. \(\quad \dfrac{4}{9}\)

  10. \(s(x)\) \(F(x)\) \(f(x)\) \(\mu_x\)
    \(1-\dfrac{x}{100}\) \(\dfrac{x}{100}\) \(\dfrac{1}{100}\) \((100-x)^{-1}{,}\)
    \(0 \leq x \leq 100\)
    \(e^{-x}{,}\)
    \(x \geq 0\)    		 \(1-e^{-x}\) \(e^{-x}\) \(1\)
    \(\dfrac{1}{1+x}\) \(1-(1+x)^{-1}{,}\)
    \(x \geq 0\)    		 \(\dfrac{1}{(1+x)^2}\) \(\dfrac{1}{1+x}\)
    \(\dfrac{(100-x)^2}{10{,}000}\) \(\dfrac{x(200-x)}{10{,}000}\) \(\dfrac{100-x}{5\,000}{,}\)
    \(0 \leq x \leq 100\)    		 \(\dfrac{2}{100-x}\)

    \(\\\)

  11. \(\quad 2.773\)

  12. \(\quad 0.00675\)

  13. \(\quad z(n-x)-z(m-x)\)

  14. \(\quad 27.55\)

  15. \(\quad\) (a) \(50\); (b) \(208.3\overline{3}\); (c) \(50\)

  16. \(\quad 0.41803\)

  17. \(x\) \(l_x\) \(p_x\) \(q_x\) \(d_x\)
    \(0\) \(100\,000\) \(0.99450\) \(0.00550\) \(550\)
    \(1\) \(99\,450\) \(0.99346\) \(0.00654\) \(650\)
    \(2\) \(98\,800\) \(0.99241\) \(0.00760\) \(750\)

\(\\\)

  1. \(x\) \(q_x\) \(l_x\) \(d_x\)
    \(0\) \(0.011\) \(1\,000\,000\) \(11\,000\)
    \(1\) \(0.005\) \(980\,000\) \(4\,945\)
    \(2\) \(0.003\) \(984\,055\) \(2\,952\)

\(\\\)

  1. \(\quad\) (a) \(0.70355\); (b) \(0.29656\); (c) \(0.02282\); (d) \(0.17768\)

  2. \(\quad 1\)

  3. \(\quad 0.75\)

  4. \(\quad 10\log{2}\)

  5. \(\quad 1.5\)

  6. \(\quad 26{,}750\)

  7. \(\quad 0.992\)

  8. \(\quad 0.4\)

  9. \(\quad\) (a) \(0.01176\); (b) \(42.5\)

  10. \(\quad 15\,000\)

  11. \(\quad 0.17324\)

  12. \(\quad 126.28\)

Answers to Life Insurance Models exercises

  1. \(\quad 5\)

  2. \(\quad 0.62073\)

  3. \(\quad 0\)

  4. \(\quad 0.04879\)

  5. \(\quad 10\)

  6. \(\quad 0.43047\)

  7. \(\quad 85\)

  8. \(\quad 0.25170\)

  9. \(\quad 17{,}442\)

  10. \(\quad 416.6\bar{6}\)

  11. \(\quad 26{,}600\)

  12. \(\quad\) (a) \(1{,}419\); (b) \(10{,}514\)

  13. \(\quad 49{,}122\)

  14. \(\quad 0.05\)

  15. \(\quad 0.76204\)

  16. \(\quad 6\)

  17. \(\quad 3{,}241.38\)

  18. \(\quad 5{,}597\)

  19. \(\quad 58{,}739\)

  20. \(\quad 111{,}612.50\)

  21. \(\quad 353.45\)

  22. \(\quad 0.96129\)

  23. \(\quad 6{,}952.38\)

  24. \(\quad 20{,}876.49\)

  25. \(\quad\) –

  26. \(\quad 0.27733\)

  27. \(\quad 0.00542\)